An Analytic Center Cutting Plane Method to Determine Complete Positivity of a Matrix

Abstract

We propose an analytic center cutting plane method to determine if a matrix is completely positive, and return a cut that separates it from the completely positive cone if not. This was stated as an open (computational) problem by Berman, D\"ur, and Shaked-Monderer [Electronic Journal of Linear Algebra, 2015]. Our method optimizes over the intersection of a ball and the copositive cone, where membership is determined by solving a mixed-integer linear program suggested by Xia, Vera, and Zuluaga [INFORMS Journal on Computing, 2018]. Thus, our algorithm can, more generally, be used to solve any copositive optimization problem, provided one knows the radius of a ball containing an optimal solution. Numerical experiments show that the number of oracle calls (matrix copositivity checks) for our implementation scales well with the matrix size, growing roughly like $O(d^2)$ for $d\times d$ matrices. The method is implemented in Julia, and available at https://github.com/rileybadenbroek/CopositiveAnalyticCenter.jl.